Friday, September 19, 2014

The Conic Sections

It is widely known that the "conic sections" are the curves of intersection of a plane with a double-napped cone (i.e., two cones placed point-to-point), as illustrated above. The three most important conic sections are the ellipse, the parabola and the hyperbola. While circles are also conic sections, they are just special cases of the ellipse.

These three famous conic sections are the "nondegenerate" conics. There are also three "degenerate" conics, formed when the plane that intersects the cones passes through the cones' point. When this is done at a steep angle, the resulting curve of intersection is a pair of crossed lines. When the plane's angle matches the angle of the cone, the plane and cones intersect along a line; when the plane's angle is shallow, the plane and cones only intersect at a single point. The animation above shows how one can transition through these six conic sections. The view on the right is of the curve of intersection when looking straight at the plane of intersection, regardless of its tilt.

The nondegenerate conics possess some incredible properties. As discussed in some of our other blog posts (see here and here), these conics have interesting, and very useful, reflective properties. (The shapes of satellite dishes are the 3D analogue of parabolas; mirrors based on ellipses and hyperbolas are useful in things like telescopes.)

One thing that gets us excited is that these conics can be defined in multiple, yet equivalent, ways. We define the conics above by intersecting a plane with a double-napped cone. (We encourage the reader to find ways to casually drop the term "double-napped cone" in a conversation.) These conics are also commonly defined by a "distance property." For instance, given a line and a point (called the "focus") that's not on the line, the set of all points that are the same distance from the line & focus form a parabola. It is really interesting (to us) to note that regardless of which definition you start with, you can derive the other as a "fact." For instance, starting with the double-napped cone definition of the parabola (see how smoothly we threw that term in there?), one can show there is a line and point in space that satisfy the "distance property" definition of the parabola. What's more, the distance formulas for each conic lend themselves to the formation of certain equations that one can use to further study their properties.

The conic sections are old; they were extensively studied by Apollonius of Perga (c. 262- c.190 B.C.), who wrote a book Conics. Much (all?) of what we commonly study about the conic sections today was developed in his book (though he did not necessarily discover all of these things). In fact, we owe the names of the conics to him. We'll leave the details for a future post, but in short, a certain construction based on a parabola lines up exactly with a certain line segment. Apollonius (and maybe others before him) gave the parabola its name as parabole means "comparison" or "application". The same construction based on an ellipse falls short of lining up with the line segment, hence the name is derived from the Greek elleipsis meaning "falling short." Finally, the construction based on a hyperbola exceeds the length of the line segment, hence its name is based on huperbole, meaning "excess" or "above". (Hence the use of the word "hyperbole" in English.)

For more information about the conic sections ... Google it.

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